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PHYSICAL REVIEW B 94, 014209 (2016)
Particlelike wave packets in complex scattering systems
Benoıt Gerardin,1 Jerome Laurent,1 Philipp Ambichl,2 Claire Prada,1 Stefan Rotter,2 and Alexandre Aubry1,*
1ESPCI Paris, PSL Research University, CNRS, Institut Langevin, UMR 7587, 1 rue Jussieu, F-75005 Paris, France2Institute for Theoretical Physics, Vienna University of Technology (TU Wien), Wiedner Hauptstraβe 8-10/136, A-1040 Vienna, Austria
(Received 24 January 2016; revised manuscript received 11 May 2016; published 25 July 2016)
A wave packet undergoes a strong spatial and temporal dispersion while propagating through a complexmedium. This wave scattering is often seen as a nightmare in wave physics whether it be for focusing, imaging,or communication purposes. Controlling wave propagation through complex systems is thus of fundamentalinterest in many areas, ranging from optics or acoustics to medical imaging or telecommunications. Here, westudy the propagation of elastic waves in a cavity and a disordered waveguide by means of laser interferometry.From the direct experimental access to the time-delay matrix of these systems, we demonstrate the existenceof particlelike wave packets that remain focused in time and space throughout their complex trajectory. Due totheir limited dispersion, their selective excitation will be crucially relevant for all applications involving selectivewave focusing and efficient information transfer through complex media.
DOI: 10.1103/PhysRevB.94.014209
I. INTRODUCTION
Waves propagating in complex media typically undergodiffraction and multiple scattering at all the inhomogeneitiesthey encounter. As a consequence, a wave packet suffersfrom strong temporal and spatial dispersion while propagatingthrough a scattering medium. Eventually, the incident waveis converted into a diffuse halo that gives rise to a com-plicated interference pattern (speckle) at the output of themedium. Albeit complex, this wave field remains, however,deterministic. By actively shaping the wave field at the input,one can manipulate the interference between all the scatteringpaths that the wave can follow. On the one hand, this insighthas given rise to spectacular focusing schemes in whichscattering enables—rather than impedes—wave focusing andpulse compression [1–9]. On the other hand, it can lead toan optimized control of wave transport [10–13]. A designedwave front can, e.g., be completely transmitted/reflected atwill [14,15] as a result of a multiple-scattering interferencethat is intrinsically narrow band [16]. Here we will aim atthe more challenging goal to generate states that are fullytransmitted/reflected, yet very robust in a broadband spectralrange. As we will demonstrate explicitly, this goal can bereached by way of highly collimated scattering states that areconcentrated along individual particlelike bouncing patternsinside the medium [17]. By avoiding the multipath interferenceassociated with conventional scattering states, these wavebeams also avoid the frequency sensitivity associated withthis interference. As we shall see, particlelike scattering statesgive rise, in the time domain, to wave packets that remainfocused in time and space throughout their trajectory withinthe medium. This crucial feature makes these states uniquelysuited for many applications in a variety of fields, rangingfrom high intensity focused ultrasound [18,19] or underwateracoustics [20,21] to endoscopic microscopy [22–24], fiberoptics [25–29], or telecommunications [30–32].
The key aspect of our experimental study is to demonstratethat these particlelike wave packets can be created just based
on the information stored in the scattering matrix [17]. Thishigh dimensional S matrix relates any arbitrary wave fieldat the input to the output of the scattering medium, and inprinciple, allows the reconstruction or prediction of either. Itfully describes wave propagation across a scattering mediumand can meanwhile be routinely measured not only in acoustics[33,34], but also in microwave technology [13,35] and optics[6,12]. The sub-blocks of the scattering matrix contain thecomplex-valued transmission (t,t′) and reflection (r, r′) matri-ces with a certain number N of input and output channels,
S =(
r t′t r′
). (1)
To describe the statistical properties of S for wave transportthrough complex media, random matrix theory (RMT) hasbeen very successful [36]. One of the striking results ofRMT is the universal bimodal distribution followed by thetransmission eigenvalues T of tt† [37] through diffusive media[36,38,39] or chaotic cavities [40,41]. In contradiction with aclassical diffusion or chaotic picture, a substantial fraction ofpropagation channels are found to be essentially closed (T ∼0) or open (T ∼ 1). Going beyond such a statistical approach,Rotter et al. [17] recently showed how a system-specificcombination of fully open or fully closed channels may lead toscattering states that follow the particlelike bouncing pattern ofa classical trajectory throughout the entire scattering process.Such particlelike scattering states with transmission close to 1or 0 are eigenstates of the Wigner-Smith time-delay matrix:
Q = − i
2πS†∂f S, (2)
where ∂f denotes the derivative with respect to the frequencyf . Originally introduced by Wigner in nuclear scatteringtheory [42] and extended by Smith to multichannel scatteringproblems [43], the Q matrix generally describes the time thatthe incoming wave accumulates due to the scattering process:each eigenvalue yields the time delay of the associatedscattering state. Compared to a mere study of the S matrix,the Q matrix provides an elegant and powerful tool to harnessthe dispersion properties of a complex medium. In this paper,we show, in particular, how a time-delay eigenstate can be
2469-9950/2016/94(1)/014209(9) 014209-1 ©2016 American Physical Society
BENOIT GERARDIN et al. PHYSICAL REVIEW B 94, 014209 (2016)
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FIG. 1. The two systems under investigation consist of (a) aregular cavity and (b) a disordered slab machined in an elastic plate.In both configurations, the S matrix is measured in the time domainbetween two arrays of points placed on the left and right sides ofthe system (see Appendix A). Flexural waves are generated on eachpoint by a pulsed laser via thermoelastic conversion over a focal spotof 1 mm2. The normal component of the plate vibration is measuredwith an interferometric optical probe. The laser source and the probeare both mounted on two-dimensional (2D) translation stages.
engineered to be “particlelike” not only in its stationary wavefunction patterns [17], but also in the sense that a nondispersivewave packet can be propagated along the correspondingparticlelike bouncing pattern. The associated eigenvalue of Qthen corresponds to the propagation time of this wave packet.
Our experimental setup consists of an elastic cavity and adisordered elastic wave guide at ultrasound frequencies (seeFig. 1). In a first step we measure the entries of the S and Q ma-trices over a large bandwidth using laser-ultrasonic techniques.The eigenvalues of the transmission matrix are shown to followthe expected bimodal distributions in both configurations.The wave fields associated with the open/closed channelsare monitored within each system in the time domain bylaser interferometry. Not surprisingly, they are shown to bestrongly dispersive as they combine various path trajectoriesand thus many interfering scattering phases. To reduce thisdispersion and to lift the degeneracy among the open/closedchannels, we consider the eigenstates of the Q matrix thathave a well-defined time delay, corresponding to a wavethat follows a single path trajectory. In transmission, a one-to-one association is found between time-delay eigenstatesand ray-path trajectories. The corresponding wave functionsare imaged in the time domain by laser interferometry. Thesynthesized wave packets are shown to follow particleliketrajectories along which the temporal spreading of the incidentpulse is minimal. In reflection, the Q matrix yields thecollimated wave fronts that focus selectively on each scattererof a multitarget medium. Contrary to alternative approachesbased on time-reversal techniques [44–47], the discrimination
FIG. 2. (a) Real part of the S matrix measured in the cavity[Fig. 1(a)] at f0 = 0.30 MHz. The black lines delimit transmissionand reflection matrices as depicted in Eq. (1). (b) Transmissioneigenvalue histogram ρ(T ) averaged over the frequency bandwidthf = 0.23–0.37 MHz. The distribution is compared to the bimodallaw ρb(T ) [red continuous line, Eq. (3)]. (c),(d) Absolute values ofthe monochromatic wave fields (f0 = 0.30 MHz) associated with thetwo most open channels.
between several targets is not based on their reflectivity but ontheir position. The eigenvalues of Q directly yield the time offlight of the pulsed echoes reflected by each scatterer.
II. EXPERIMENTAL RESULTS
A. Revealing the open and closed channels in a cavity
The waves we excite and measure are flexural waves ina duralumin plate of dimension 500 × 40 × 0.5 mm3 (seeFig. 1). The frequency range of interest spans from 0.23 to0.37 MHz (�f = 0.14 MHz) with a corresponding averagewavelength λ of 3.5 mm. We thus have access to N = 2W/λ ∼22 independent channels, W being the width of the elasticplate. Two complex scattering systems are built from thehomogeneous plate: (i) a regular cavity formed by cutting theplate over 20 mm on both sides of the plate [see Fig. 1(a)] and(ii) a scattering slab obtained by drilling several circular holesin the plate [see Fig. 1(b)]. The thickness L of each system is45 and 52 mm, respectively.
The S matrix is measured for each system with thelaser-ultrasonic setup described in Fig. 1, following theprocedure explained in the Appendix A. Transmission andreflection matrices are expressed in the basis of the modesof the homogeneous plate [15]. These eigenmodes and theireigenfrequencies have been determined theoretically using thethin elastic plate theory [48,49]. They are renormalized suchthat each of them carries unit energy flux across the platesection [15]. Figure 2(a) displays an example of an S matrix
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recorded at the central frequency f0 = 0.30 MHz for thecavity. Most of the energy emerges along the main diagonal andtwo subdiagonals [50] of the reflection/transmission matrices.These reflection and transmission matrix elements correspondto specular reflection of each mode on the cavity boundariesand to the ballistic transmission of the incident wave front,respectively.
We first focus on the statistics of the transmission eigenval-ues Tl computed from the measured t matrix (see Appendix B).Their distribution ρ(T ) is estimated by averaging the corre-sponding histograms over the frequency bandwidth. Figure2(b) shows the comparison between the distribution measuredin the rectangular cavity and the bimodal law ρb which istheoretically expected in the chaotic regime [40,41],
ρb(T ) = 1
π√
T (1 − T ). (3)
Even though our system is not chaotic, but exhibits regulardynamics, a good agreement is found between the measuredeigenvalue distribution and the RMT predictions, confirmingprevious numerical studies [51]. A similar bimodal distributionof transmission eigenvalues is obtained in the disordered plate,as shown in the Supplemental Material [52].
Whereas the eigenvalues Tl of tt† yield the transmissioncoefficients of each eigenchannel, the corresponding eigenvec-tors provide the combination of incident modes that allow us toexcite this specific channel. Hence, the wave field associatedwith each eigenchannel can be measured by propagating thecorresponding eigenvector. To that aim, the whole systemis scanned with the interferometric optical probe. A set ofimpulse responses is measured between the line of sources atthe input and a grid of points that maps the medium. The wavefunction associated with a scattering state is then deducedby a coherent superposition of these responses weighted bythe amplitude and phase of the eigenvector at the input (seeAppendix C). Hence all the wave functions displayed in thispaper are only composed of experimentally measured dataand do not imply any theoretical calculation or numericalsimulation.
Figures 2(c) and 2(d) display the wave field associatedwith the two most open eigenchannels (Tl ∼ 1) of the cavity.Although such open channels allow a full transmission of theincident energy, they do not show a clear correspondence witha particular path trajectory. The same observation holds inthe disordered plate [52]. As a consequence, the associatedscattering state undergoes multiple scattering when passingthrough the cavity. Figures 3(a) and 3(b) illustrate thisdispersion by displaying the output temporal signal associatedwith the two open channels shown in Figs. 2(c) and 2(d) (seeAppendix B). Both signals contain several peaks occurring ataltogether three different times of flight. As we will see further,each peak is associated with a particular path trajectory andcan be addressed independently by means of the Wigner-Smithtime-delay matrix.
B. Addressing particlelike scattering states in a cavity
The Wigner-Smith time-delay matrix Q is now investigatedto generate coherent scattering states from the set of openchannels. Since Q is Hermitian when derived from a unitary
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S matrix [Eq. (2)], the time-delay eigenstates qinm form an
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In general, qinm is a 2N -dimensional eigenvector which implies
an injection from both the left and the right leads of the system.However, among this set of time-delay eigenstates, a subsetfeatures an incoming flux from only one lead that also exitsthrough just one of the leads. These are exactly the desiredstates that belong to the subspace of open or closed channelsand display trajectorylike wave-function patterns.
As was shown by Rotter et al. [17], the above arguments canbe translated into a straightforward operational procedure (seeAppendix B), which we apply here to identify the particlelikescattering states among the time-delay eigenstates of themeasured Q matrix. The litmus test for this procedure inthe present context will be to show that the three differenttime traces that are identifiable in the open transmissionchannels [see Figs. 3(a) and 3(b)] can now be individuallyaddressed through an associated particlelike state. The resultswe obtain for the cavity geometry [Fig. 1(a)] fully confirm oursuccessful implementation of particlelike scattering states: Thepropagation of the states we obtain from our procedure yieldsmonochromatic wave states that are clearly concentrated onindividual bouncing patterns (see Fig. 4). Whereas Fig. 4(a)
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corresponds to the direct path between the input and outputleads, Figs. 4(b) and 4(c) display a more complex trajectorywith two and four reflections on the boundaries of the cavity,respectively. The associated time delays τm do correspond tothe run times of a particle that would follow the same trajectoryat the group velocity vg ∼ 2.6 mm/μs of the flexural wave[53]. Their transmission coefficients |tm| are equal to 0.90,0.95, and 0.85, respectively, meaning that they are almost fullytransmitted through the cavity.
The time trace associated with each particlelike scatteringstate is computed from the frequency-dependent t matrix (seeAppendix B). The result is displayed in Figs. 3(c)–3(e). Unlikethe open transmission channels studied above, each particlelikestate gives rise to a single pulse that arrives at the outputtemporally unscattered at time τ = τm. Figure 3 also showsthat each temporal peak in the time trace of the open channelscan be attributed to a particular path trajectory. We may thusconclude that the open channel displayed in Fig. 2(c) is mainlyassociated with the double and quadruple scattering pathsdisplayed in Figs. 4(b) and 4(c). The open channel displayed inFig. 2(d) consists of a linear combination of the paths displayedin Fig. 4. This association is also confirmed by explicitlyanalyzing the vectorial decomposition of the particlelike statein terms of the transmission eigenchannel basis.
The frequency dependence of the particlelike scatteringstates is investigated in the Supplemental Material [52].They are shown to be stable over the frequency ranges f =0.2–0.6 MHz [Fig. 4(a)], f = 0.3–0.6 MHz [Fig. 4(b)], andf = 0.2–0.4 MHz [Fig. 4(c)]. The corresponding bandwidthsare at least one order of magnitude larger than the frequency
correlation width of the transmission matrix coefficients whichis equal to 0.02 MHz [52]. This proves the robustness ofparticlelike states over a broadband spectral range. Given thisnondispersive feature, they turn out to be perfect candidatesalso for the formation of minimally dispersive wave packetsin the time domain. To check this conjecture, we investigatehere the spatiotemporal wave functions of these states over theaforementioned bandwidths (see Appendix B). The propaga-tion of the particlelike wave packets through the cavity canbe visualized in the three first movies of the SupplementalMaterial [52]. Figure 5 displays successive snapshots of thewave packet synthesized from the particlelike scattering statedisplayed in Fig. 4(c). Quite remarkably, the spatiotemporalfocus of the incident wave packet is maintained throughout itstrajectory despite the multiple-scattering events it undergoesin the cavity.
C. Lifting the degeneracy of particlelike scattering states
In a next step, we investigate particlelike scattering statesin the disordered wave guide [Fig. 1(b)]. The corresponding Qmatrix is measured at the central frequency f0 = 0.3 MHz (seeAppendix A). Figures 6(a) and 6(b) display the monochromaticwave functions associated with two fully transmitted time-delay eigenstates. Whereas Fig. 6(a) displays the typicalfeatures of a particlelike scattering state that winds its waythrough the scatterers, the time-delay eigenstate of Fig. 6(b)is clearly associated with two scattering paths of identicallength. We thus encounter here a degeneracy in the time-delay eigenvalues that needs to be lifted by an additionalcriterion, such as by considering well-defined subspaces ofthe measured S matrix [54]. In this instance, the two ray pathscan be discriminated by their different angles of incidence.Correspondingly, we consider two subspaces S′ of the originalS matrix by keeping either positive or negative angles of inci-dence from the left lead (see Appendix D). The correspondingtime-delay matrices lead to two distinct particlelike scatteringstates displayed in Figs. 6(c) and 6(d). The two scatteringpaths that were previously mixed in the original time-delayeigenstate [Fig. 6(b)] are now clearly separated. The frequencystability of these states is investigated in the SupplementalMaterial [52]. They are shown to be stable over the frequencyranges f = 0.2–0.4 MHz [Fig. 6(c)] and f = 0.2–0.5 MHz[Fig. 6(d)]. The corresponding particlelike wave packets areshown in the two last movies of the Supplemental Material[52] where the high quality of their focus in space and time isimmediately apparent.
D. Revealing time-delay eigenstates in reflection
Time-delay eigenstates can also result from a suitablecombination of closed channels. Figures 6(e) and 6(f) displaytwo such closed channels derived from the S matrix ofthe disordered slab. The closed channels combine multiplereflections from the holes of the scattering layer. However, theclosed channel displayed in Fig. 6(e) mixes the contributionsfrom the scatterers labeled 1 and 2. Also the closed channeldisplayed in Fig. 6(f) is associated with reflections fromaltogether three scatterers (2, 3, and 4). A simple eigenvaluedecomposition of rr† or tt† does not allow a discrimination
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FIG. 6. Absolute value of the wave field associated with several time-delay/scattering eigenstates of the disordered system at the centralfrequency f0. (a) Transmitted particlelike scattering state (τm = 29 μs, |tm| = 0.84). (b) Degenerate time-delay eigenstate mixing two particleliketrajectories (τm = 26 μs, |tm| = 0.91). (c),(d) Unmixing of the time-delay eigenstate displayed in (b) by considering different subspaces of theS matrix (see Appendix D). Their transmission coefficients |tm| are equal to 0.94 and 0.98, respectively. (e),(f) Closed channels deduced fromthe S matrix (Tl ∼ 0) showing simultaneous reflections from the scatterers of the disordered wave guide. (g) Reflected time-delay eigenstateshowing selective focusing on scatterer 2 (time delay τm = 11 μs, reflection coefficient |rm| = 0.94). (h) Reflected time-delay eigenstateshowing selective focusing on scatterer 3 (time delay τm = 30 μs, reflection coefficient |rm| = 0.88).
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between the scatterers. On the contrary, the analysis of the Qmatrix allows a one-to-one correspondence with each scattererbased on a time-of-flight discrimination. Figures 6(g) and 6(h)actually display the wave functions associated with tworeflected time-delay eigenstates. Each of these eigenstates isassociated with a reflection from a single scatterer (2 and 3,respectively). The corresponding time delays τm given in thecaption of Fig. 6 are directly related to the depth z of eachscatterer, such that τm ∼ 2z/vg .
III. DISCUSSION
The first point we would like to emphasize is the relevanceof the time-delay matrix for selective focusing and imaging inmultitarget media. The state-of-the-art approach is the DORTmethod (French acronym for Decomposition of the TimeReversal Operator). Initially developed for ultrasound [44,45]and more recently extended to optics [46,47], this widelyused approach takes advantage of the reflection matrix r tofocus iteratively by time-reversal processing on each scattererof a multitarget medium. Mathematically, the time-reversalinvariants can be deduced from the eigenvalue decompositionof the time reversal operator rr† or, equivalently, from thesingular value decomposition of r. On the one hand, theeigenvectors of r should, in principle, allow selective focusingand imaging of each scatterer. On the other hand, the associatedeigenvalue directly yields the scatterer reflectivity. However, aone-to-one association between each eigenstate of r and eachscatterer only exists under a single-scattering approximationand if the scatterers exhibit sufficiently different reflectivities.Figures 6(e) and 6(f) illustrate this limit. Because the holesdisplay similar scattering cross sections in the disordered slab,closed channels are associated with several scatterers at once.On the contrary, the time-delay matrix allows a discriminationbetween scatterers based on the time-of-flight of the reflectedechoes [see Figs. 6(g) and 6(h)]. Moreover, unlike the DORTmethod, a time-delay analysis also allows us to discriminatethe single-scattering paths from multiple-scattering events, thelatter ones corresponding to longer time of flight. Hence,the time-delay matrix provides an alternative and promis-ing route for selective focusing and imaging in multitargetmedia.
A second relevant point to discuss is the nature of transmit-ted time-delay eigenstates in other complex systems. Recently,Carpenter et al. [27] and Xiong et al. [28] investigated thegroup delay operator, i/(2π )t−1∂f t, in a multimode opticalfiber. The eigenstates of this operator are known as theprincipal modes in fiber optics [25,26]. For a principal modeinput, a pulse that is sufficiently narrow band reaches the outputtemporally undistorted, although it may have been stronglyscattered and dispersed along the length of propagation. Onthe contrary, for a particlelike scattering state input, thefocus of the pulse is not only retrieved at the output of thescattering medium but maintained throughout its entire propa-gation through the system. This crucial difference providesparticlelike states with a much broader frequency stabilitythan principal modes, which translates into the possibility tosend much shorter pulses through these particlelike scatteringchannels. Last but not least, we also emphasize that eventhough particlelike states are unlikely to occur in diffusive
scattering media, the time-delay eigenstates are still veryrelevant also in such a strongly disordered context. Considerhere, e.g., that the eigenstates with the longest time delay canbe of interest for energy storage, coherent absorption [55], orlasing [56,57] purposes. From a more fundamental point ofview, the trace of the Q matrix directly provides the density ofstates of the scattering medium [58], a quantity that turns outto be entirely independent of the mean free path in a disorderedsystem [59].
Finally, we would like to stress the impact of our studyon other fields of wave physics and its extension to morecomplex geometries. For this experimental proof of concept,a measurement of the wave field inside the medium wasrequired in order to image the wave functions and prove theirparticlelike feature. However, such a sophisticated protocol isnot needed to physically address particlelike wave packets.A simple measurement of the scattering matrix (or a subpartof it) at neighboring frequencies [27] yields the time-delaymatrix from which particlelike state inputs can be extracted.Such a measurement can be routinely performed through3D scattering media whether it be in optics [6,12], in themicrowave regime [13,35], or in acoustics [33,34]. As tothe generation of particlelike wave packets, multielementtechnology is a powerful tool for the coherent control ofacoustic waves and electromagnetic waves [60]. Moreover,recent progress in optical manipulation techniques now allowsfor a precise spatial and temporal control of light at the inputof a complex medium [60]. Hence there is no obstacle for theexperimental implementation of particlelike wave packets inother fields of wave physics. At last, we would like to stressthe fact that in our experimental implementation we are in thelimit of only a few participating modes with a wavelength thatis comparable to the spatial scales of the system. In this limitthe implementation of particlelike states is truly nontrivialsince interference and diffraction dominates the scatteringprocess as a whole. When transferring the concept to theoptical domain one may easily reach the geometric opticslimit where the wavelength is much shorter than most spatialscales of the system and particlelike states may in fact bemuch easier to implement in corresponding complex opticalmedia.
IV. CONCLUSION
In summary, we experimentally implemented particlelikescattering states in complex scattering systems. Based onan experimentally determined time-delay matrix, we havedemonstrated the existence of wave packets that followparticlelike bouncing patterns in transmission through or inreflection from a complex scattering landscape. Strikingly,these wave packets have been shown to remain focused intime and space throughout their trajectory within the medium.We are convinced that the superior properties of these statesin terms of frequency stability and spatial focus will makethem very attractive for many applications of wave physics,ranging from focusing to imaging or communication purposes.In transmission, the efficiency of these states in terms ofinformation transfer as well as their focused input and outputprofile will be relevant. In reflection, selective focusing basedon a time-of-flight discrimination will be a powerful tool to
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overcome aberration and multiple scattering in detection andimaging problems.
ACKNOWLEDGMENTS
The authors wish to thank A. Derode for fruitful discussionsand advice. The authors are grateful for funding providedby LABEX WIFI (Laboratory of Excellence within theFrench Program Investments for the Future, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*). B.G. acknowledgesfinancial support from the French “Direction Generale del’Armement”(DGA). P.A. and S.R. were supported by theAustrian Science Fund (FWF) through Projects NextLiteF49-10 and I 1142-N27 (GePartWave).
APPENDIX A: EXPERIMENTAL PROCEDURE
The first step of the experiment consists of measuring theimpulse responses between two arrays of points placed on theleft and right sides of the disordered slab (see Fig. 1). Thesetwo arrays are placed 5 mm away from the disordered slab.The array pitch is 0.8 mm (i.e., <λ/2) which guarantees asatisfying spatial sampling of the wave field. Flexural wavesare generated in the thermoelastic regime by a pumped diodeNd:YAG laser (THALES Diva II) providing pulses havinga 20-ns duration and 2.5 mJ of energy. The out-of-planecomponent of the local vibration of the plate is measuredwith a heterodyne interferometer. This probe is sensitive tothe phase shift along the path of the optical probe beam.The calibration factor for mechanical displacement normalto the surface (100 mV/nm) was constant over the detectionbandwidth (100–400 kHz). Signals detected by the opticalprobe were fed into a digital sampling oscilloscope andtransferred to a computer. The impulse responses betweeneach point of the same array (left and right) form the time-dependent reflection matrices (r and r′, respectively). Theset of impulse responses between the two arrays yields thetime-dependent transmission matrices t (from left to right)and t′ (from right to left). From these four matrices, onecan build the S matrix in a point-to-point basis [Eq. (1)]. Adiscrete Fourier transform (DFT) of S is then performed overa time range �t = 120 μs that excludes the echoes due toreflections on the ends of the plate and ensures that most of theenergy has escaped from the sample when the measurement isstopped. The next step of the experimental procedure consistsof decomposing the S matrices in the basis of the flexuralmodes of the homogeneous plate. These eigenmodes and theireigenfrequencies have been determined theoretically usingthe thin elastic plate theory [15,48,49]. They are normalizedsuch that each of them carries unit energy flux across theplate section. Theoretically, energy conservation would implythat S is unitary. In other words, its eigenvalues should bedistributed along the unit circle in the complex plane. However,as shown in a previous work [15], this unitarity is not retrievedexperimentally because of experimental noise. A dispersionof the eigenvalues si of the S matrix is observed aroundthe unit circle. We compensate for this undesirable effectby considering a normalized scattering matrix with the sameeigenspaces but with normalized eigenvalues [15]. The Qmatrix is then deduced from S using Eq. (2). The frequency
derivative of S at f = f0 is estimated from the centered finitedifference,
∂f S(f0) = S(f0 + δf ) − S(f0 − δf )
2δf,
with δf = 3 kHz.
APPENDIX B: REVEALINGTRANSMISSION/TIME-DELAY EIGENCHANNELSAND THEIR TEMPORAL/SPECTRAL FEATURES
The transmission and time-delay eigenchannels are derivedfrom the matrices S and Q measured at the central frequencyf0. The transmission matrix t(f0) (from the left to the rightlead) is extracted from S(f0) [Eq. (1)]. The output and inputtransmission eigenvectors, ul and vl , are derived from thesingular value decomposition of t(f0):
t(f0) =∑
l
√Tl(f0)ul(f0)v†l (f0)
with Tl(f0) the intensity transmission coefficient associatedwith the lth scattering eigenstate at the central frequency. Thefrequency-dependent amplitude transmission coefficient tl(f )of this eigenstate can be obtained from the set of transmissionmatrices t(f ) measured over the whole frequency bandwidth,such that
tl(f ) = u†l (f0)t(f )vl(f0).
An inverse DFT of tl(f ) finally yields the time-dependentamplitude transmission coefficient tl(τ ) of the lth scatteringeigenstate measured at the central frequency f0. The timetraces displayed in Figs. 3(a) and 3(b) correspond to the squarenorm of this quantity.
The time-delay eigenchannels at the central frequency arederived from the eigenvalue decomposition of Q(f0)
Q(f0) =∑m
τmqinm (f0)
[qin
m (f0)]†
.
The time-delay eigenvector qinm is a 2N -dimensional column
vector that can be decomposed as
qin(f0) =(
qinm,L(f0)
qinm,R(f0)
),
where qinm,L(f0) and qin
m,R(f0) contain the complex coefficientsof qin
m (f0) in the basis of the N incoming modes in the leftand right leads, respectively. Among the set of time-delayeigenstates, particlelike scattering states injected from the leftlead should fulfill the following condition [17]:∣∣∣∣qin
m,L(f0)∣∣∣∣2 � ∣∣∣∣qin
m,R(f0)∣∣∣∣2 � 0.
A particlelike scattering state is thus associated with a N -dimensional input eigenvector qin
m,L(f0). The correspondingoutput eigenvector qout
m,R(f0) and transmission coefficient
t(q)m (f0) can be deduced from the t matrix:
t(f0)qinm,L(f0) = t (q)
m (f0)qoutm,R(f0).
The frequency-dependent amplitude transmission coefficientt
(q)m (f ) of this time-delay eigenstate can be obtained from the
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set of transmission matrices t(f ) measured over the wholefrequency bandwidth, such that
t (q)m (f ) = [
qoutm,L(f0)
]†t(f )qin
m,R(f0).
An inverse DFT of t(q)m (f ) finally yields the time-dependent
amplitude transmission coefficient t(q)m (τ ) of the mth time-
delay eigenstate measured at the central frequency f0. The timetraces displayed in Figs. 3(c)–3(e) correspond to the squarenorm of this quantity.
APPENDIX C: IMAGING SPATIOTEMPORALWAVE FUNCTIONS OF TRANSMISSION/REFLECTION
AND TIME-DELAY EIGENCHANNELS
Impulse responses are measured between the line of sources(denoted by the index i) and a grid of points (denoted by theindex j ) that maps the medium, following the same procedureas the one described above. The grid pitch is 1.3 mm. Thisset of impulse responses forms a transmission matrix k(τ ) =[kji(τ )]. A discrete Fourier transform (DFT) of k(τ ) yields a setof frequency-dependent transmission matrices k(f ). The linesof k(f ) are then decomposed in the basis of the plate modes.The monochromatic wave field (f ) = [ψj (f )] associatedwith a transmission/reflection or time-delay eigenchannel isprovided by the product between the matrix k(f ) and thecorresponding eigenvector [vl(f0) or qin
m,L(f0), respectively].
The time-dependent wave field (τ ) = [ψj (τ )] is deduced byan inverse DFT over a frequency bandwidth of our choice.Note that a Hann window function is priorly applied to k(f )to limit side lobes in the time domain.
APPENDIX D: UNMIXING DEGENERATEDTIME-DELAY EIGENSTATES
Depending on the geometry of the scattering medium, thedifferent scattering paths involved in a degenerated time-delayeigenstate can be discriminated either in the real space or in thespatial frequency domain [54]. Here, the time-delay eigenstatedisplayed in Fig. 6(b) shows two scattering paths with oppositeangles of incidence. Hence, they can be discriminated byanalyzing each block of the S matrix in the spatial frequencydomain. The left lead of each block is decomposed over thepositive or negative angles of incidence. This subspace of theS matrix, referred to as S′, is then used to compute a reducedtime-delay matrix Q′ [54] such that
Q′ = − i
2πS′−1
∂f S′.
Note that the transpose conjugate operation of Eq. (2) is herereplaced by an inversion of S′ because of its nonunitarity [54].Depending on the sign of the angle of incidence chosen forthe left lead, the reduced matrix Q′ provides the time-delayeigenstates displayed in Figs. 6(c) and 6(d).
[1] A. Derode, P. Roux, and M. Fink, Robust Acoustic TimeReversal with High-Order Multiple Scattering, Phys. Rev. Lett.75, 4206 (1995).
[2] M. Tanter, J.-L. Thomas, and M. Fink, Time reversal and theinverse filter, J. Acoust. Soc. Am. 108, 223 (2000).
[3] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo,and M. Fink, Time Reversal of Electromagnetic Waves, Phys.Rev. Lett. 92, 193904 (2004).
[4] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Focusingbeyond the diffraction limit with far-field time reversal, Science315, 1120 (2007).
[5] I. M. Vellekoop and A. P. Mosk, Focusing coherent light throughopaque strongly scattering media, Opt. Lett. 32, 2309 (2007).
[6] S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara,and S. Gigan, Measuring the Transmission Matrix in Optics: AnApproach to the Study and Control of Light Propagation inDisordered Media, Phys. Rev. Lett. 104, 100601 (2010).
[7] J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A.Lagendijk, Control Of Light Transmission Through OpaqueScattering Media in Space and Time, Phys. Rev. Lett. 106,103901 (2011).
[8] D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A.Walmsley, S. Gigan, and B. Chatel, Spatio-temporal focusingof an ultrafast pulse through a multiply scattering medium, Nat.Commun. 2, 447 (2011).
[9] O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Focusingand compression of ultrashort pulses through scattering media,Nat. Photon. 5, 371 (2011).
[10] I. M. Vellekoop and A. P. Mosk, Universal Optimal Transmissionof Light Through Disordered Materials, Phys. Rev. Lett. 101,120601 (2008).
[11] J. B. Pendry, Light finds a way through the maze, Physics 1, 20(2008).
[12] M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, andW. Choi, Maximal energy transport through disordered mediawith the implementation of transmission eigenchannels, Nat.Photon. 6, 583 (2012).
[13] Z. Shi and A. Z. Genack, Transmission Eigenvalues and the BareConductance in the Crossover to Anderson Localization, Phys.Rev. Lett. 108, 043901 (2012).
[14] W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, Transmissioneigenchannels in a disordered medium, Phys. Rev. B 83, 134207(2011).
[15] B. Gerardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, FullTransmission and Reflection of Waves Propagating Through aMaze of Disorder, Phys. Rev. Lett. 113, 173901 (2014).
[16] C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, andH. Cao, Broadband Coherent Enhancement of Transmission andAbsorption in Disordered Media, Phys. Rev. Lett. 115, 223901(2015).
[17] S. Rotter, P. Ambichl, and F. Libisch, Generating ParticlelikeScattering States in Wave Transport, Phys. Rev. Lett. 106,120602 (2011).
[18] J.-F. Aubry, M. Pernot, F. Marquet, M. Tanter, and M. Fink,Transcostal high-intensity-focused ultrasound: ex vivo adap-tive focusing feasibility study, Phys. Med. Biol. 53, 2937(2008).
[19] E. Cochard, C. Prada, J.-F. Aubry, and M. Fink, Ultrasonicfocusing through the ribs using the dort method, Med. Phys. 36,3495 (2009).
[20] G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A.Kuperman, and H. C. Song, An initial demonstration of
014209-8
PARTICLELIKE WAVE PACKETS IN COMPLEX . . . PHYSICAL REVIEW B 94, 014209 (2016)
underwater acoustic communication using time reversal, IEEEJ. Ocean. Eng. 27, 602 (2002).
[21] C. Prada, J. de Rosny, D. Clorennec, J.-G. Minonzio,A. Aubry, M. Fink, L. Berniere, P. Billand, S. Hibral, andT. Folegot, Experimental detection and focusing in shallowwater by decomposition of the time reversal operator, J. Acoust.Soc. Am. 122, 761 (2007).
[22] T. Cizmar and K. Dholakia, Exploiting multimode waveguidesfor pure fibre-based imaging, Nat. Commun. 3, 1027 (2012).
[23] I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, Focusingand scanning light through a multimode optical fiber usingdigital phase conjugation, Opt. Express 20, 10583 (2012).
[24] M. Ploschner, T. Tyc, and T. Cizmar, Seeing through chaos inmultimode fibres, Nat. Photon. 9, 529 (2015).
[25] S. Fan and J. M. Kahn, Principal modes in multimode waveg-uides, Opt. Lett. 30, 135 (2005).
[26] A. A. Juarez, C. A. Bunge, S. Warm, and K. Petermann,Perspectives of principal mode transmission in mode-division-multiplex operation, Opt. Express 20, 13810 (2012).
[27] J. Carpenter, B. J. Eggleton, and J. Schroder, Observation ofeisenbud–wigner–smith states as principal modes in multimodefibre, Nat. Photon. 9, 751 (2015).
[28] W. Xiong, P. Ambichl, Y. Bromberg, S. Rotter, and H. Cao,Spatio-temporal control of light transmission through a mul-timode fiber with strong mode coupling, arXiv:1601.04646[Phys. Rev. Lett. (to be published)].
[29] B. C. Carpenter, J. Thomsen and T. D. Wilkinson, Degeneratemode-group division multiplexing, J. Lightw. Technol. 30, 3946(2012).
[30] J. Salz, Digital transmission over cross-coupled linear channels,AT&T Tech. J. 64, 1147 (1985).
[31] G. G. Raleigh and J. M. Cioffi, Spatio-temporal coding forwireless communication, IEEE Trans. Commun. 46, 357 (1998).
[32] A. Tulino and S. Verdu, Random matrix theory and wirelesscommunications, Found. Trends Commun. Inform. Theory 1, 1(2004).
[33] R. Sprik, A. Tourin, J. de Rosny, and M. Fink, Eigenvaluedistributions of correlated multichannel transfer matrices instrongly scattering systems, Phys. Rev. B 78, 012202 (2008).
[34] A. Aubry and A. Derode, Random Matrix Theory Applied toAcoustic Backscattering and Imaging In Complex Media, Phys.Rev. Lett. 102, 084301 (2009).
[35] O. Dietz, H.-J. Stockmann, U. Kuhl, F. M. Izrailev, N. M.Makarov, J. Doppler, F. Libisch, and S. Rotter, Surface scatteringand band gaps in rough waveguides and nanowires, Phys. Rev.B 86, 201106 (2012).
[36] C. W. J. Beenakker, Random-matrix theory of quantum trans-port, Rev. Mod. Phys. 69, 731 (1997).
[37] The symbol † here stands for transpose conjugate.[38] O. N. Dorokhov, On the coexistence of localized and extended
electronic states in the metallic phase, Solid State Commun. 51,381 (1984).
[39] Y. Imry, Active transmission channels and universal conduc-tance fluctuations, Europhys. Lett. 1, 249 (1986).
[40] H. U. Baranger and P. A. Mello, Mesoscopic Transport ThroughChaotic Cavities: A Random S-Matrix Theory Approach, Phys.Rev. Lett. 73, 142 (1994).
[41] R. A. Jalabert, J.-L. Pichard, and C. W. J. Beenakker, Universalquantum signatures of chaos in ballistic transport, Europhys.Lett. 27, 255 (1994).
[42] E. P. Wigner, Lower limit for the energy derivative of thescattering phase shift, Phys. Rev. 98, 145 (1955).
[43] F. T. Smith, Lifetime matrix in collision theory, Phys. Rev. 118,349 (1960).
[44] C. Prada and M. Fink, Eigenmodes of the time reversal operator:A solution to selective focusing in multiple-target media, WaveMotion 20, 151 (1994).
[45] C. Prada, S. Manneville, D. Spoliansky, and M. Fink, Decom-position of the time reversal operator: Detection and selectivefocusing on two scatterers, J. Acoust. Soc. Am. 99, 2067(1996).
[46] S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, andS. Gigan, Exploiting the Time-Reversal Operator for AdaptiveOptics, Selective Focusing, and Scattering Pattern Analysis,Phys. Rev. Lett. 107, 263901 (2011).
[47] A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, andA. Aubry, Smart optical coherence tomography for ultra-deepimaging through highly scattering media, arXiv:1510.08613.
[48] M. C. Cross and R. Lifshitz, Elastic wave transmission at anabrupt junction in a thin plate with application to heat transportand vibrations in mesoscopic systems, Phys. Rev. B 64, 085324(2001).
[49] D. H. Santamore and M. C. Cross, Surface scattering analysisof phonon transport in the quantum limit using an elastic model,Phys. Rev. B 66, 144302 (2002).
[50] The more energetic subdiagonals of the transmission/reflectionmatrices account for the conversion between even and oddmodes of similar momentum.
[51] F. Aigner, S. Rotter, and J. Burgdorfer, Shot Noise in theChaotic-To-Regular Crossover Regime, Phys. Rev. Lett. 94,216801 (2005).
[52] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.94.014209 for the scattering matrix analysisof the disordered waveguide, the frequency study of particlelikescattering states and five movies showing particlelike wavepackets propagating through the cavity and the disorderedwaveguide.
[53] E. Dieulesaint and D. Royer, Elastic Waves in Solids I (Springer-Verlag, Berlin, 2000).
[54] A. Brandstotter, Master’s thesis, Institute for TheoreticalPhysics, Vienna University of Technology, 2016.
[55] Y. D. Chong and A. D. Stone, Hidden Black: CoherentEnhancement of Absorption in Strongly Scattering Media, Phys.Rev. Lett. 107, 163901 (2011).
[56] N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, TamingRandom Lasers Through Active Spatial Control of the Pump,Phys. Rev. Lett. 109, 033903 (2012).
[57] N. Bachelard, S. Gigan, X. Noblin, and P. Sebbah, Adaptivepumping for spectral control of random lasers, Nat. Phys. 10,426 (2014).
[58] M. Davy, Z. Shi, J. Wang, X. Cheng, and A. Z. Genack,Transmission Eigenchannels and the Densities of States ofRandom Media, Phys. Rev. Lett. 114, 033901 (2015).
[59] R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati,and S. Rotter, Invariance property of wave scattering throughdisordered media, Proc. Natl. Acad. Sci. USA 111, 17765(2014).
[60] A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Controllingwaves in space and time for imaging and focusing in complexmedia, Nat. Photon. 6, 283 (2012).
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